Unit 3093: Linear Models and Regression Analysis

1. Introduction to Linear Models

• Definition of linear relationships
• Independent vs. dependent variables
• Purpose and applications of linear models in data analysis

2. Simple Linear Regression

• Concept and formulation of simple linear regression
• Fitting a linear equation: y = β0 + β1x + ε
• Interpretation of regression coefficients
• Prediction using simple linear regression

3. Multiple Linear Regression

• Extension from simple to multiple independent variables
• Model formulation: y = β0 + β1x1 + β2x2 + ... + βnxn + ε
• Interpretation of coefficients in multiple regression
• Use cases and examples

4. Assumptions of Linear Regression

• Linearity: relationship between predictors and response
• Independence of errors
• Homoscedasticity: constant variance of errors
• Normality of residuals
• Methods to test assumptions (scatterplots, Durbin-Watson test, residual plots, Q-Q plots)

5. Assessing Model Fit

• Coefficient of determination (R-squared)
• Adjusted R-squared for multiple predictors
• Statistical significance of regression coefficients (t-tests)
• F-test for overall model significance

6. Residual Analysis

• Definition and calculation of residuals
• Interpreting residual plots
• Detecting outliers and influential points
• Validity checks for regression model

7. Variable Selection and Model Building

• Importance of selecting relevant variables
• Stepwise regression (forward selection and backward elimination)
• Criteria for variable inclusion/exclusion (p-values, AIC, BIC)
• Building robust and parsimonious models

8. Collinearity in Linear Models

• Definition and causes of collinearity
• Effects on coefficient estimates and model stability
• Detecting collinearity (Variance Inflation Factor - VIF, condition indices)
• Remedies: variable removal, combining variables, principal component regression

9. Polynomial Regression

• Extending linear models to capture nonlinear relationships
• Model formulation with polynomial terms (e.g., quadratic, cubic)
• Interpretation and visualization of polynomial regression
• Overfitting and model complexity considerations

10. Practical Applications of Linear Models

• Economics: forecasting and trend analysis
• Social sciences: studying relationships between social variables
• Healthcare: predicting patient outcomes
• Engineering: quality control and process optimization
• Case studies demonstrating prediction and decision-making using linear models